January 19

Exercise 0 (8.3 (b),(c),(d),(e)).

Find all incongruent solutions to each of the following congruences:

(b)
$6x\equiv 5 (\mod 7)$
(c)
$x^2 \equiv 1 (\mod 8)$
(d)
$x^2 \equiv 2 (\mod 7)$
(e)
$x^2 \equiv 3 (\mod 7)$

Question 1 (12.2 (a)).

Show that there are infinitely many primes of the for 6n+5.

Exercise 2.

Determine the value of
$2^{2012}+3^{3013} (\mod 7)$

Question 3.

Show that the equation
$x^{10}+y^{10}=11z^{10}$
has no non-zero integer solutions.

Does it have non-zero solutions mod 6?

Question 4 (based on 2.1 (b)).

Show that in a primitive Pythagorean triple a,b,c exactly one of a,b,c is divisible by 5.

Question 5 (8.4 (d))

Show that a number is divisible by 9 if and only if its sum of digits is divisible by 9.